The one, most recommended treatment for small injuries is simple, and
is just like treating a small 'human' cut: using Neosporin* ointment!
(*the proprietary name for the triple antibiotic ointment containing neomycin,
bacitracin zinc and polymyxin B)

It has been used it with considerable success in treating superficial
injuries and infections. It stays on well under wet conditions. The generic
product is much cheaper (Triple Antibiotic Ointment). It is effective
against gram-negative bacteria, which predominate in most foggy infections.
Just make sure the affected area is dry before applying to get the best
result.

But remember, even the smallest of cuts can get easily infected in such
warm moist conditions. And make sure you keep the frog extra clean
if he has an injury.